Perversions and whips: Static and dynamic problems of elastic filaments

Abstract

Two problems of elastic filaments are considered, one a problem of determining the static shape of a filament with intrinsic curvature tinder constant force, and one a problem of determining the dynamical behavior of a planar rod. In the first problem, examining the phenomenon of perversion, methods of dynamical systems are used to examine the static equations of elastic filaments, in which the arc-length of the filament plays the role of time. The phenomenon of perversion, in which two oppositely handed helices are connected by an inversion of chirality, is represented by a heteroclinic orbit of the dynamical system. The second problem is an examination of a whip wave, the propagation of a loop in a whip as it travels the length of the whip to create a sharp crack as the loop reaches the end of the rod and accelerates to supersonic speeds. This study is undertaken in two stages: first we examine the propagation of the loop as it travels down the rod far from the end of the rod, and then we examine the behavior of the rod as the loop reaches the end of the rod and unfolds, accelerating the tip. In the first stage we use techniques of asymptotic analysis and perturbation methods to determine the relationship of the speed of the loop to the radius of the rod. In the second stage we employ a numerical technique to compute the behavior of the loop as it unfolds to determine the relationship of the maximal speed of the tip of the rod to the characteristics of the rod and the forces applied to the handle.